The Academia Sectrorum Naturae, the first scientific community, was founded in
1560, and in order to be included in membership, you had to discover a new law of nature, a prerequisite for admission.A. 1560, and in order to be included in membership, you had to discover a new law of nature, a prerequisite for admission
Wrong. Parallelism problem: "and" is parallel marker, but "was founded in 1950" and "in order to be included in membership" are NOT parallel.
B. 1560, and included a prerequisite for admission: in order to be included, you had to have made the discovery of a new law of nature
Wrong. The usage of colon
":" means you're explaining for something ==> Present simple tense is enough. But in B, "you had to" means past tense. Hence, B is wrong.
C. 1560 with a prerequisite for admission: the discovery of a new law of nature
Correct. "with a prerequisite ...." just modifies the subject "The Academia Sectrorum Nature". Does not change meaning.
After a colon ":", you can use noun/noun phrase/clause. Because the part after the colon is just an explanation for something before the colon.
Other example: Medication X was initiated in 1990 with the main purpose: helping human being live longer.
D. the year 1560, including the prerequisite that to be admitted, you needed to have discovered a new law of nature
Wrong. "the year 1560" is wordy, "in 1560" is enough.
" including the prerequisite that to be admitted" is not grammatical;
"you needed to have discovered" is bad grammar.
E. 1560 with the rule that in order to be admitted for membership, you had to discover a new law of nature, that was the prerequisite
Wrong. "you had" means past tense. But the sentence just wants to explain a rule in general, so present simple tense is better.
"that" refers to "a new law of nature" incorrectly.
Hope it helps.